Comparison of the Bergman kernel and the Carath\'eodory--Eisenman volume

TL;DR

The paper establishes that the Carathéodory–Eisenman volume and the Bergman kernel are quantitatively comparable to the volume of the Carathéodory metric's indicatrix in complex domains, confirming a conjecture in several complex variables.

Contribution

It proves the comparability of the Carathéodory–Eisenman volume with the Bergman kernel volume, extending the multidimensional Suita conjecture to all domains in complex space.

Findings

Carathéodory–Eisenman volume is comparable to the Carathéodory metric indicatrix volume.

The Bergman kernel volume is also comparable to these volumes via the Suita conjecture.

Constants depend only on the dimension n.

Abstract

It is proved that for any domain in $\mathbb C^n$ the Caratheodory--Eisenman volume is comparable with the volume of the indicatrix of the Caratheodory metric up to small/large constants depending only on $n.$ Then the "multidimensional Suita conjecture" theorem of Blocki and Zwonek implies a comparable relationship between these volumes and the Bergman kernel.

Equations

$$C{E}_D(z)=\sup \{|\det F^{\prime }(z){|}^2:F\in \mathcal{O}(D,{\mathbb{D}}^n)\}.$$

$$g_D(z,w)=\sup \{u(w):u\in \text{mbox}PSH(D),u<0,u(\zeta )<\log ||\zeta -z||+C\},$$

$$A_D(z;X)=\underset{\lambda \to 0}{\mathrm{limsup}}\frac{\exp (g_D(z,z+\lambda X))}{|\lambda |},$$

$$C_D(z;X)=\sup \{|f^{\prime }(z)X|:f\in \mathcal{O}(D,\mathbb{D})\}.$$

$$K_D(z)=\sup \{|f(z){|}^2:f\in L_h^2(D),||f|{|}_{L^2(D)}\le 1\}.$$

$${\delta }_D(z;X):=\sup \{r>0:z+\lambda X\in D\text{mbox}if|\lambda |<r\}.$$

$${\delta }_D^{-1}(z;\mu X)=|\mu |{\delta }_D^{-1}(z;X),\mu \in \mathbb{C},$$

$$I_D(z)=\{X\in {\mathbb{C}}^n:{\delta }_D^{-1}(z;X)<1\}.$$

$$V_D\le V{A}_D\le V{C}_D.$$

$$K_D(z)\le (V_D(z){)}^{-1}.$$

$$K_D\ge (V{A}_D{)}^{-1}.$$

$$c^{-2n}(V{A}_D{)}^{-1}\ge (V_D{)}^{-1}\ge K_D\ge (V{A}_D{)}^{-1}.$$

$$C_n(V{C}_D{)}^{-1}\ge C{E}_D\ge c_n(V{C}_D{)}^{-1}.$$

$$C_n{K}_D\ge C{E}_D.$$

$$C_n{K}_D\ge C{E}_D\ge c^{2n}{c}_n{K}_D.$$

$$D\supset D_{c,r}:=\{z\in r{\mathbb{D}}^2:\text{mbox}Re{z}_1+(\text{mbox}Re{z}_2{)}^2<c(\text{mbox}Im{z}_2{)}^2\},c>1,r>0.$$

$${\kappa }_G(z;X)=\inf \{|\alpha |:\exists \phi \in \mathcal{O}(\mathbb{D},G):\phi (0)=z,\alpha {\phi }^{\prime }(0)=X\}.$$

$$\underset{\delta \to 0+}{\mathrm{limsup}}{\delta }^{3/4}{\kappa }_{D_{c,r}}((-\delta ,0);e_1)<\infty .$$

$$C_D(a;X)=C_{D_a}(a;X)={\delta }_{D_a}^{-1}(a;X).$$

$${\delta }_D(a;e_j)=\text{mbox}dist\left(a,a+\text{mbox}Span(e_k,k\ge j)\setminus D\right).$$

$$k_n\sum _{j=1}^n|X_j|r_j(a)\le C_D(a;X)\le \sum _{j=1}^n|X_j|r_j(a),$$

$$|\det F^{\prime }(a)|\ge (k_n{)}^n{\Pi }_D(a).$$

$$\left|\det {\left(\frac{\partial f_i}{\partial z_j}(a)\right)}_{\begin{array}{c}1\le i\le m\\ 1\le j\le m\end{array}}\right|\ge (k_n{)}^m\prod _{j=1}^m{r}_j(a).$$

$$V_j:=(-1{)}^{m+1+j}\det {\left(\frac{\partial f_i}{\partial z_l}(a)\right)}_{\begin{array}{c}1\le i\le m\\ 1\le l\le m+1,l\ne j\end{array}},1\le j\le m+1.$$

$$\det {\left(\frac{\partial f_i}{\partial z_j}(a)\right)}_{\begin{array}{c}1\le i\le m+1\\ 1\le j\le m+1\end{array}}=\sum _{j=1}^{m+1}{V}_j\frac{\partial f_{m+1}}{\partial z_j}(a)=f_{m+1}^{\prime }(a)V.$$

$$\begin{gathered} C_D(a;V)\ge k_n\sum _{j=1}^{m+1}|V_j|r_j(a)\ge k_n|V_{m+1}|r_{m+1}(a) \\ =k_n{r}_{m+1}(a)\left|\det {\left(\frac{\partial f_i}{\partial z_j}(a)\right)}_{\begin{array}{c}1\le i\le m\\ 1\le j\le m\end{array}}\right|\ge (k_n{)}^{m+1}\prod _{j=1}^{m+1}{r}_j(a). \end{gathered}$$

$$|\det F^{\prime }(a)|\le n!{\Pi }_D(a).$$

$$(k_n{)}^{2n}\le C{E}_D(a).({\Pi }_D(a){)}^{-2}\le (n!{)}^2.$$

$$E_a:=\{z\in {\mathbb{C}}^n:\sum _{j=1}^n{r}_j(a)|z_j-a_j|<1\},$$

$$k_n^{2n}V{C}_D(a)\le \text{mbox}Vol(E_a)=\frac{(2\pi {)}^n}{(2n)!({\Pi }_D(a){)}^2}\le V{C}_D(a).$$

Full text

The first named author is partially supported by the Bulgarian National Science Fund, Ministry of Education and Science of Bulgaria under contract DN 12/2. This paper was started while his was visiting the Paul Sabatier University, Toulouse in November 2018 as a guest professor.

In recent years, the interest in holomorphically invariant objects has grown from quantities stemming from maps to or from the one-dimensional disc to quantities related to the $n$-dimensional ball. The main focus of interest has been the squeezing function, which measures how big the one-to-one image of a domain can be while remaining inside the unit ball (and sending a base point to the origin of the ball).

The Carathéodory–Eisenman “volume” is a variant on that idea, at the infinitesimal level. Let $\mathbb{D}\subset\mathbb{C}$ be the unit disc. Given $D$ be a domain in $\mathbb{C}^{n},$ and $z\in D$,

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We are using the polydisc $\mathbb{D}^{n}$ for technical reasons. Replacing it by the unit ball in $\mathbb{C}^{n},$ we get the same function up to small/large constants independent of $D$.

Unfortunately, the lack of a higher-dimensional analogue to the Koebe quarter theorem prevents us from relating our results to the squeezing function, but the behaviour of $CE_{D}(z)$ can be related to some basic geometric objects associated to the domain. We need more definitions.

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Let $L_{h}^{2}(D)$ be the Bergman space of $D$, i.e. the Hilbert space of all square-integrable holomorphic functions $f$ on $D$. Let $K_{D}$ be the restriction to the diagonal of the Bergman kernel of $D$. Recall that

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Denote by $\delta_{D}(z;X)$ the distance from $z$ to $\partial D$ along the vector $X$:

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Observe that $\delta^{-1}_{D}(z;X)$ is $1$-homogeneous in $X$, that is,

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so it is this quantity that we want to compare to the various infinitesimal metrics that occur in complex analysis.

Let $I_{D}(z)$ be the indicatrix of $\delta^{-1}_{D}(z;\cdot),$ that is,

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Then $z+I_{D}(z)$ is the maximal balanced subdomain of $D$ centered at $z.$

Set $IA_{D}$ and $IC_{D}$ to be the indicatrices of $A_{D}$ and $C_{D}.$ Note that $IC_{D}(z)$ is a convex set.

Since $C_{D}\leq A_{D}\leq\delta^{-1}_{D},$ then $I_{D}\subset IA_{D}\subset IC_{D}$ and hence

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On the other hand, if $G$ is a balanced domain, then $(1,f)_{L_{2}(G)}=0$ for any $f\in L_{2}(G)$ with $f(0)=0.$ Therefore $K_{G}(0)=(\mbox{Vol}\ G)^{-1}$. In particular, applying this to $G=z+I_{D}(z)\subset D$,

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The following opposite inequality is called the multidimensional Suita conjecture (see [1, Theorem 7.5] and [2, Theorem 2]).

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Note that if $D$ is $\mathbb{C}$-convex, resp. convex, then $A_{D}\geq C_{D}\geq c\delta^{-1}_{D}$ with $c=1/4$, resp. $c=1/2$ (see e.g. [5, Proposition 1], resp. the remark after this proposition). Thus, Corollary 4 applies to those cases and we reobtain $K_{D}\leq c^{-2n}(VA_{D})^{-1}$ as in [2, Theorem 5].

The aim of this note is to prove a version of Theorem 3 for $CE_{D},$ comparing it to the volume of the Carathéodory indicatrix (see Definition 2).

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In particular, if $D$ is pseudoconvex, then by Theorem 3 and (1),

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Proof of Corollaries 4 and 6. We only have to show that under the assumption $A_{D}\geq c\delta^{-1}_{D}$ (resp. $C_{D}\geq c\delta^{-1}_{D}$), $D$ is pseudoconvex. Suppose it is not. By [3, Theorem 4.1.25], after an affine change of coordinates, we may suppose that $0\in\partial D$ and

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Recall that the Kobayashi-Royden metric of a domain $G$ in $\mathbb{C}^{n}$ is given by

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It follows by the proof of [4, Theorem 1.1] that

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Then the inequalities $C_{D}\leq A_{D}\leq\kappa_{D}\leq\kappa_{D_{c,r}}$ lead to a contradiction.

Proof of Theorem 5. Let $a\in D.$ Note that $D_{a}:=IC_{D}(a)$ is a convex balanced domain centered at $a,$ and hence

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The proof of [5, Proposition 14] rests on the construction, in a $\mathbb{C}$-convex domain $D$, of an orthonormal basis $e_{1},\dots,e_{n}$ of $\mathbb{C}^{n}$ such that for $1\leq j\leq n$,

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Since $D_{a}$ is convex, using (2), we deduce from [5, (4)] that one may find a constant $k_{n}>0$ depending only on $n$ such that

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where the $X_{j}$ are the coordinates of $X$ in the basis $e_{1},\dots,e_{n}$ and $r_{j}(a)=C_{D}(a;e_{j}).$

Let $\Pi_{D}(a)=\prod_{j=1}^{n}r_{j}(a)$.

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Then define $V\in\mathbb{C}^{m+1}$ to be the vector of cofactors

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Choose $f_{m+1}\in\mathcal{O}(D,\mathbb{D})$ to be extremal for the Carathéodory metric in the $V$ direction, so that $|f_{m+1}^{\prime}(a)V|=C_{D}(a;V)$. By our choice of $V$,

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By (3) and the recursion assumption,

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∎

On the other hand, from the definition of the Carathéodory metric, for any map $F\in\mathcal{O}(D,\mathbb{D}^{n})$ one has that

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It follows that

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Now we compare $\Pi_{D}(a)^{-2}$ with $VC_{D}(a)$. Define the diamond domain

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the inequalities (3) imply that $k_{n}D_{a}\subset E_{a}\subset D_{a}$, and so

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Combining (4) and (5), we get that

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$\Box$

Remark. Let $P_{D}(a)=1/\min\Pi_{D}(a),$ where the minimum is taken over all orthonormal bases of $\mathbb{C}^{n}.$ Denote by $V_{D}^{i}(a)$ and $V_{D}^{e}(a)$ the maximal volume of a polydisc $\Delta_{a}$ centered at $a$ such that $\Delta_{a}\subset D_{a},$ respectively the minimal volume of a polydisc $\Delta_{a}$ centered at $a$ such that $\Delta_{a}\supset D_{a}.$ It follows from the proof above that the functions $V_{D}^{i},V_{D}^{e},VE_{D},P_{D}$ and $(CE_{D})^{-1}$ are equal up to small/large constants depending only on $n.$